Since then, the government has tightened its rules for oil payments. But with natural gas, the Bush administration recently loosened the rules and eased its audits intended to uncover cheating.

Industry executives deny any wrongdoing, arguing that the disparities stem primarily from different rules for calculating the sale prices for paying royalties and the sale prices for informing shareholders.

"The price of gas downstream is always going to be higher because you have costs that have to be recouped for getting it to the customer," said Robert H. Davis, a spokesman for Exxon Mobil. "You have to process the gas. You have to transport it, and you have to sell it. There will always be a discrepancy there."

Companies that pump oil and gas on federal property are required to pay the government royalties, usually 12 percent to 16 percent of the value of what they sell.

Royalties for natural gas have climbed sharply in the last three years. But while prices nearly doubled from 2001 to 2005, the $5.15 billion in gas royalties for 2005 was less than the $5.35 billion in 2001. When oil and gas are combined, royalties were about $8 billion in 2005, almost the same as in 2001.

Because much of the information about specific transactions is kept secret, it remains unclear to what extent, if at all, the weakness in royalty payments stems from deliberate cheating or from issues with the rules themselves.

But one major producer, Burlington Resources, admitted to shareholders last year that it might have underpaid about $76 million in gas royalties in the 1990's. And in Alabama, a jury ruled in 2003 that Exxon had cheated on $63.6 million worth of royalties from gas wells in state-owned waters. The jury awarded $11.9 billion in punitive damages, which a judge later reduced to $3.5 billion. Exxon disputes the charges and is appealing the verdict.

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The possible losses to taxpayers in gas could be even higher than the losses tied to the scandals over oil royalties. For one thing, natural gas production on federal land is worth twice as much as oil.

Moreover, the Interior Department has scaled back on full audits, pushed out a couple of its more aggressive auditors and been criticized by its own inspector general for the audits that it did pursue.

"We are talking about the same issues and in many cases the same players as before," said Danielle Brian, executive director of the Project on Government Oversight, a nonprofit watchdog group that exposed many of the oil royalty scandals.

"These companies had knowingly been cheating on oil for years, if not decades," Ms. Brian continued. "To ignore the likelihood that the same thing is happening on the gas side is absurd."

Johnnie M. Burton, director of the Interior Department's Minerals Management Service, said the disparities were mostly the result of deductions that the regulations let companies take, reducing the sale prices they report to the government.

But Ms. Burton said she had not known and could not explain why companies were reporting higher sale prices to their shareholders and to the Securities and Exchange Commission than to her office.

"I can't answer because I don't know," she said in an interview. "We don't look at S.E.C. filings. We don't have enough staff to do all of that. If we were to do that, then we would have to have more staff and more budget. You know, there is such a thing as budget constraint, and it's been real tough, let me tell you." The contrasts between what companies are telling the government and what they are telling shareholders is stark.

The Interior Department, using the numbers given by companies paying royalties, said the average sale price of natural gas on federal leases was $5.62 per thousand cubic feet in fiscal 2005, which ended Sept. 30.

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By contrast, Exxon told shareholders that it received about $6.88 per thousand cubic feet in the nine months that ended Sept. 30. Chevron said its average price in that period was $6.49. Kerr-McGee, which suffered huge losses from hedging against a drop in prices, nonetheless said it still received an average price of $6.59.

"There's no reason why what the companies report to their shareholders should be higher than what they report" to the Minerals Management Service, said Lee Helfrich, a lawyer who has represented California in many battles with the industry over royalties. "The ultimate goals or mission of the S.E.C. and the M.M.S. are different, but the information reported to each should be the same."

In the scandals over oil royalties in the 1990's, government investigators, aided by industry whistle-blowers and investigation by the Project on Government Oversight, found that companies were using a host of tricks to understate their sale prices.

These included buy-sell agreements in which producers swapped oil with each other at artificially low prices and then resold it at higher prices. Companies also sold oil at below-market prices to their own affiliates, classified high-priced "sweet" oil as much cheaper "sour" oil and padded their deductions for transportation costs.

In the wake of the scandals, the outgoing Clinton administration pushed through tough new rules for valuing crude oil, which relied on comparing company reports with an index of spot market prices.

A Pro-Industry Approach

But the Bush administration did not close any loopholes for valuing natural gas. Indeed, in March 2005 it expanded the list of deductions and decided against valuing sales at spot-market prices when companies were selling to their own affiliates.

The industry-friendly stance was intentional. Mr. Bush and top White House officials also placed a top priority on promoting domestic energy production. Vice President Dick Cheney's energy task force called for giving lucrative new incentives to companies that drill in the Gulf of Mexico and other high-risk areas.

The Bush administration also took a much more relaxed approach to auditing and fraud prevention. In 2003, the Interior Department's inspector general declared that the auditing process was "ineffective" and "lacked accountability" and that many of the auditors were unqualified.

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In one instance, inspectors discovered that auditors had lost the working papers for an important audit and tried to cover up their blunder by creating and back-dating false documents. Rather than punish anybody, the inspector general recounted, the minerals service gave the employee who produced the new documents a financial bonus for "creativity."

Administration officials said last week that they had addressed most of the criticisms and that the inspector general had since said its corrective actions were "sufficient."

The Interior Department also fired two of its most aggressive and successful auditors. One of them was Bobby L. Maxwell, a veteran auditor who had recovered hundreds of millions of dollars in underpayments over a 22-year career and received an award for meritorious service in 2003 from Interior Secretary Gale A. Norton.

Mr. Maxwell was fired in early 2005 after clashing with superiors over his belief that Kerr-McGee had shortchanged the government $12 million. Mr. Maxwell charged that he had been wrongfully fired, and the government paid him an undisclosed amount of money to settle out of court.

Mr. Maxwell is now pursuing Kerr-McGee, which has denied any guilt, with his own lawsuit under the False Claims Act, which allows private citizens who prove fraud to collect some of the money they help recover.

Patrick Etchart, a spokesman for the Minerals Management Service in Denver, said that Mr. Maxwell lost his job because of a reorganization and that he had declined an offer to move to a different city.

But lawmakers who wrestled with the government over previous royalty scandals are dubious.

"It's all gotten worse, not better," said Representative Carolyn B. Maloney, Democrat of New York, who led Congressional investigations into cheating on oil royalties in the 1990's. "They make the process so complicated that no one can really follow the money."

Ending Detailed Inspections

Perhaps the most striking example of sluggish auditing is the government's effort to collect back royalties from companies that blatantly ignored one of the government's basic rules.

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Under current rules aimed at promoting energy production in deep waters, companies can produce large volumes of oil and gas without paying royalties at all. But the rules also require companies to start paying royalties if market prices climb above certain "threshold" levels.

As it happens, market prices have been above those levels since the 2003 fiscal year. But even though dozens of companies never bothered to start paying, Ms. Burton said earlier this month that the government had yet to demand repayment three months into the 2006 fiscal year.

"It's more complicated than you might think," said Lucy Querques Dennett , associate director of the Minerals Management Service in charge of the issue.

But enforcing the rules about price thresholds is easy compared with verifying the actual sale value of natural gas.

Over the last four years, the Bush administration has ordered its auditors to move away from detailed inspections in favor of a more cursory approach of looking for anomalies in company reports. If a company in Louisiana, say, reported prices that differed from those of other companies in the same region, it would attract closer scrutiny.

Mr. Etchart, the agency's spokesman, said that the number of full-scale audits had declined slightly over the past few years and that the budget for compliance had fallen.

But he said the government still took a "close look" at 71 percent of oil and gas production. "Our strategy would obviously be to focus on anomalies," he said, "but it is also to focus on large producing areas."

The agency's strategy has drawn protests, however, from many states, which are entitled to a share of federal royalties, and from some of the Interior Department's most aggressive auditors.

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One of those auditors was Kevin Gambrell, director of the Federal Indian Minerals Office in Farmington, N.M. Mr. Gambrell fought with his superiors over many issues, one of which was their demand that he do fewer audits and simply monitor posted prices of companies in the same area.

"Where the M.M.S. approach falls short is that there are so many different types of deductions you can take in getting gas and oil to the market, and there are so many premiums and bonuses in the contracts," Mr. Gambrell said in a recent interview. "You have to take a detailed look at the contracts to know what's going on."

The Interior Department forced Mr. Gambrell out in 2003, charging that he had improperly destroyed office documents. Mr. Gambrell sued for wrongful termination, arguing that he had discarded only copies of documents. He also presented evidence that his office had recovered eight times as much money as offices that used the administration's preferred approach.

The government settled his case in 2004 by clearing him of any wrongdoing and paying him an undisclosed amount of money.

For practical purposes, the biggest cost to taxpayers may have less to do with cheating and fraud than with the government's inscrutable rules.

Consider the case of Burlington Resources, a Houston-based producer that ConocoPhillips acquired in December for $35.6 billion. Burlington paid $8.5 million in 2001 to settle charges of cheating related to its oil royalties. Last March, Burlington disclosed that it might also have underpaid gas royalties by about $76 million during the 1990's. It set aside $81 million to cover possible litigation costs.

Unlike others, Burlington executives provided information to The Times on the royalties it paid for natural gas and on the sale prices that it has reported to the Interior Department since 2002.

During those four years, Burlington said it paid $627 million in gas royalties and that its annual payment shot up from $89 million in 2002 to $233 million in 2005.

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That surge in royalties does track closely with the rise in market prices. But Burlington's numbers also highlight the essential issue raised by many critics: the rules let companies understate the value of their gas sales by taking scores of deductions.

Those deductions include the cost of transportation, processing, brokerage fees, pipeline reservation fees and even certain "theoretical losses" for companies that own their own pipelines.

In 2001, Burlington reported an average price of $1.98 per thousand cubic feet to the government but an average sale price of $3.20 to its shareholders. In 2005, the company reported an average sale price of $5.75 to the government and $6.46 to shareholders.

Keeping Royalties Secret

James Bartlett, a spokesman for Burlington, said part of the discrepancy resulted from the fact that much of Burlington's production is in the Rocky Mountains, where natural gas fetches lower prices.

The federal government does not require companies to divulge the amount of royalties they pay or what they tell the government about sale prices. And unlike Burlington Resources, Exxon and most other major oil companies refused to disclose the information when asked.

"It's not required information," said Mr. Davis of Exxon, echoing responses from Chevron, Royal Dutch/Shell and other big producers. "We're not going to publish it."